Question

6. An application of the sampling distribution of the sample proportion

Of the 21.4 million U.S. firms without paid employees, 32% are female owned. [Data source: U.S. Census Bureau; data based on the 2007 Economic Census.]

A simple random sample of 408 firms is selected. Use the Distributions tool to help you answer the questions that follow.

The probability that the sample proportion is within ±.01 of the population proportion is:

a. 0.5000

b. 0.3328

c. 0.0160

d. 0.1664

Suppose the sample size is increased to 808. The probability that the sample proportion is within ±.01 of the population proportion is now:

a. 0.2357

b. 0.0160

c. 0.4714

d. 0.4992

Answer 1

a)

Here, μ = 0.32, σ = 0.0231, x1 = 0.31 and x2 = 0.33. We need to compute P(0.31<= X <= 0.33). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.31 - 0.32)/0.0231 = -0.43
z2 = (0.33 - 0.32)/0.0231 = 0.43

Therefore, we get
P(0.31 <= X <= 0.33) = P((0.33 - 0.32)/0.0231) <= z <= (0.33 - 0.32)/0.0231)
= P(-0.43 <= z <= 0.43) = P(z <= 0.43) - P(z <= -0.43)
= 0.6664 - 0.3336
= 0.3328

b)
Here, μ = 0.32, σ = 0.016, x1 = 0.31 and x2 = 0.33. We need to compute P(0.31<= X <= 0.33). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.31 - 0.32)/0.016 = -0.63
z2 = (0.33 - 0.32)/0.016 = 0.63

Therefore, we get
P(0.31 <= X <= 0.33) = P((0.33 - 0.32)/0.016) <= z <= (0.33 - 0.32)/0.016)
= P(-0.63 <= z <= 0.63) = P(z <= 0.63) - P(z <= -0.63)
= 0.7357 - 0.2643
= 0.4714